The present invention relates to an apparatus for detecting a voltage peak value and a current peak value. More particularly, the invention relates to an apparatus for detecting the amount of time and phase angle deviation between a voltage peak point and a current peak point for use in a digital processing type fault locator in which analog electrical data is converted into digital electrical data. The fault located is used to locate a faulty point on a transmission line, i.e. to determine the distance of a trouble point from a substation or the like.
If, in a conventional fault point locating method, the resistance between a fault point and the location of a fault locator position is represented by r, the inductance therebetween by l, the fault current flowing towards the fault point from a power source by i, and the present voltage at the position of the fault locator (hereinafter referred to as "a locator point" when applicable) by v, and if the ground charging current is disregarded because it is much smaller than the fault current, then the above-described factors can be related to one another by the following relation: ##EQU1##
The right side equation (1) has a term necessitating differential calculation; that is, it must be subjected to numeral differentiation when the equation is numerically evaluated. Accordingly, if the input changes abruptly, the calculation error is generally high. Therefore, in general, data represented by the following equation (2), which is obtained by integrating equation (1) with respect to time t is employed: ##EQU2##
This will be described with reference to a digital processing type locator. FIG. 1 shows a waveform which illustrates a principle of evaluating the expression (2) with digital data. Specifically, FIG. 1 indicates a procedure of numeral integration with respect to a voltage waveform by way of example. In FIG. 1, reference character h designates a sampling time width. The waveform is represented by: EQU v=V sin (.omega.t+.theta.) (3)
The instantaneous value of the expression (3) sampled at a time instant t is represented by: EQU v.sub.n =V sin (.omega.t+.theta.) (4)
Instantaneous values sampled one and two sampling times before the instantaneous value v.sub.n are represented respectively by: EQU v.sub.n-1 =V sin {.omega.(t-h)+.theta.}, and (5) EQU v.sub.n-2 =V sin {.omega.(t-2h)+.theta.}. (6)
If in the expression (2) EQU t.sub.2 =t and t.sub.1 =t-h, (7)
then it is evident that the figure ABCD, which is a part of the sinusoidal waveform, corresponds to the left side of the expression (2). Therefore, when the value h is sufficiently small, applying the trapezoidal formula, the area can be approximated as follows: ##EQU3##
By similarly approximating equation (2), the following expression (9) and (10) can be obtained for the current: ##EQU4##
In general, the following expression (11) corresponding to the expression (10) is obtained by shifting the sampling time: ##EQU5## From the expressions (10) and (11), the following expression (12) is obtained: ##EQU6##
There are serious drawbacks accompanying a conventional locator including:
(1) During the occurrence of a fault in the system, the voltage leads the fault current i by about 90.degree. (practically, the former will not lead the latter about 90.degree. because of the resistance at the fault point);
(2) The effect of a quantization error cannot be neglected because digital data is used;
(3) For relatively short transmission lines, upon occurrence of transmission line troubles, in general, the current tends to increase while the voltage tends to decrease; and
(4) If one of the data points used for numerically evaluating the expression (12) is near the peak point thereof, the other data point will be near the zero crossing point. This leads to an increase in the quantization error which is turn lowers the overall accuracy of the fault locator system.
In view of the foregoing, an object of the invention is to provide a method of decreasing the effect of a quantization error in a digital processing type fault locator by improving the calculation method.
The expressions (4) through (6) represent instantaneous values as described above. If these data values are converted into digital data by an A/D (analog-to-digital) converter, typically the expression (4) can be rewritten as follows: EQU v.sub.n =v{sin (.omega.t+.theta.)+(.epsilon./V)}, (13)
where .epsilon. is the resolution of the A/D converter which determines the quantization error. Once the dynamic range and the number of bits of the A/D converter are determined, the resolution .epsilon. determining the quantization error is determined as a constant. For instance, in the employment of an A/D converter having a dynamic range of .+-.10 V and having twelve bits including one sign bit, the resolution per bit is: ##EQU7## If an input current of 200.sqroot.2 A and a voltage of 110.sqroot.2 V are scaled to a dynamic range of 10 V, resolutions .epsilon..sub.I and .epsilon..sub.V determining quantization errors with respect to the current and the voltage are: EQU .epsilon..sub.I =69.1 mA, and (15) EQU .epsilon..sub.V =38.0 mV. (16)
Thus, quantization errors of about 5% and about 3%, respectively, are present for a current having a peak value of .sqroot.2 A and a voltage having a peak value .sqroot.2 V.
As is apparent from the expression (13), when the inputs I and V are both low, the errors and steps of resolutions with respect to the amplitudes of the inputs are high. Furthermore, even if with respect to one of the voltage V and the current A the data value corresponding to a time instant t-h is at a peak, .theta. is about 90.degree., and therefore the other data value is near the zero value. Thus, the effect of the relative error cannot be neglected.